MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mircgrextend Structured version   Visualization version   GIF version

Theorem mircgrextend 28460
Description: Link congruence over a pair of mirror points. cf tgcgrextend 28263. (Contributed by Thierry Arnoux, 4-Oct-2020.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Baseβ€˜πΊ)
mirval.d βˆ’ = (distβ€˜πΊ)
mirval.i 𝐼 = (Itvβ€˜πΊ)
mirval.l 𝐿 = (LineGβ€˜πΊ)
mirval.s 𝑆 = (pInvGβ€˜πΊ)
mirval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
mirtrcgr.e ∼ = (cgrGβ€˜πΊ)
mirtrcgr.m 𝑀 = (π‘†β€˜π΅)
mirtrcgr.n 𝑁 = (π‘†β€˜π‘Œ)
mirtrcgr.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
mirtrcgr.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
mirtrcgr.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
mirtrcgr.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
mircgrextend.1 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ))
Assertion
Ref Expression
mircgrextend (πœ‘ β†’ (𝐴 βˆ’ (π‘€β€˜π΄)) = (𝑋 βˆ’ (π‘β€˜π‘‹)))

Proof of Theorem mircgrextend
StepHypRef Expression
1 mirval.p . 2 𝑃 = (Baseβ€˜πΊ)
2 mirval.d . 2 βˆ’ = (distβ€˜πΊ)
3 mirval.i . 2 𝐼 = (Itvβ€˜πΊ)
4 mirval.g . 2 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 mirtrcgr.a . 2 (πœ‘ β†’ 𝐴 ∈ 𝑃)
6 mirtrcgr.b . 2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
7 mirval.l . . 3 𝐿 = (LineGβ€˜πΊ)
8 mirval.s . . 3 𝑆 = (pInvGβ€˜πΊ)
9 mirtrcgr.m . . 3 𝑀 = (π‘†β€˜π΅)
101, 2, 3, 7, 8, 4, 6, 9, 5mircl 28439 . 2 (πœ‘ β†’ (π‘€β€˜π΄) ∈ 𝑃)
11 mirtrcgr.x . 2 (πœ‘ β†’ 𝑋 ∈ 𝑃)
12 mirtrcgr.y . 2 (πœ‘ β†’ π‘Œ ∈ 𝑃)
13 mirtrcgr.n . . 3 𝑁 = (π‘†β€˜π‘Œ)
141, 2, 3, 7, 8, 4, 12, 13, 11mircl 28439 . 2 (πœ‘ β†’ (π‘β€˜π‘‹) ∈ 𝑃)
151, 2, 3, 7, 8, 4, 6, 9, 5mirbtwn 28436 . . 3 (πœ‘ β†’ 𝐡 ∈ ((π‘€β€˜π΄)𝐼𝐴))
161, 2, 3, 4, 10, 6, 5, 15tgbtwncom 28266 . 2 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼(π‘€β€˜π΄)))
171, 2, 3, 7, 8, 4, 12, 13, 11mirbtwn 28436 . . 3 (πœ‘ β†’ π‘Œ ∈ ((π‘β€˜π‘‹)𝐼𝑋))
181, 2, 3, 4, 14, 12, 11, 17tgbtwncom 28266 . 2 (πœ‘ β†’ π‘Œ ∈ (𝑋𝐼(π‘β€˜π‘‹)))
19 mircgrextend.1 . 2 (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ))
201, 2, 3, 4, 5, 6, 11, 12, 19tgcgrcomlr 28258 . . 3 (πœ‘ β†’ (𝐡 βˆ’ 𝐴) = (π‘Œ βˆ’ 𝑋))
211, 2, 3, 7, 8, 4, 6, 9, 5mircgr 28435 . . 3 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜π΄)) = (𝐡 βˆ’ 𝐴))
221, 2, 3, 7, 8, 4, 12, 13, 11mircgr 28435 . . 3 (πœ‘ β†’ (π‘Œ βˆ’ (π‘β€˜π‘‹)) = (π‘Œ βˆ’ 𝑋))
2320, 21, 223eqtr4d 2777 . 2 (πœ‘ β†’ (𝐡 βˆ’ (π‘€β€˜π΄)) = (π‘Œ βˆ’ (π‘β€˜π‘‹)))
241, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23tgcgrextend 28263 1 (πœ‘ β†’ (𝐴 βˆ’ (π‘€β€˜π΄)) = (𝑋 βˆ’ (π‘β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  β€˜cfv 6542  (class class class)co 7414  Basecbs 17165  distcds 17227  TarskiGcstrkg 28205  Itvcitv 28211  LineGclng 28212  cgrGccgrg 28288  pInvGcmir 28430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-trkgc 28226  df-trkgb 28227  df-trkgcb 28228  df-trkg 28231  df-mir 28431
This theorem is referenced by:  mirtrcgr  28461
  Copyright terms: Public domain W3C validator